Problem introduced in Definition REF  is thus a Kantorovich problem with an extra entropic penalty \((\pi |\otimes )\) . This entropy is computed with respect to the input marginals to ease the discussion, but other reference measures can be used (such as in [1]}).
As \(\epsilon \)  increases, the approximation of the original OT problem degrades, but as detailed in Section REF , the computational scheme converges faster, and it is more stable in high dimension.
To derive a simple computational scheme, we detail below the dual formulation, which is obtained through Fenchel-Rockafellar duality theorem [2]}, and corresponds to the optimization of a pair of continuous functions \((f,g)\in ()^2\) .
